Computing a spanning tree (ST) and a minimum spanning tree (MST) of a graph are fundamental problems in Graph Theory and arise as a subproblem in many applications. In this paper, we propose parallel algorithms to these problems. One of the steps of previous parallel MST algorithms relies on the heavy use of parallel list ranking which, though efficient in theory, is very time-consuming in practice. Using a different approach with a graph decomposition, we devised new parallel algorithms that do not make use of the list ranking procedure. We proved that our algorithms are correct, and for a graph G = (V;E), |V| = n and |E| = m, the algorithms can be executed on a BSP/CGM model using O(log p) communications rounds with O((n+m)/p) computation time for each round. To show that our algorithms have good performance on real parallel machines, we have implemented them on GPU (Graphics Processing Unit). The obtained speedups are competitive and showed that the BSP/CGM model is suitable for designing general purpose parallel algorithms.